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Properties

Label 2-1911-1911.635-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.976 - 0.216i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.149 − 0.988i)3-s + (−0.433 − 0.900i)4-s + (0.294 − 0.955i)7-s + (−0.955 + 0.294i)9-s + (−0.826 + 0.563i)12-s + (−0.997 + 0.0747i)13-s + (−0.623 + 0.781i)16-s + (−1.70 − 0.457i)19-s + (−0.988 − 0.149i)21-s + (−0.294 − 0.955i)25-s + (0.433 + 0.900i)27-s + (−0.988 + 0.149i)28-s + (1.82 + 0.488i)31-s + (0.680 + 0.733i)36-s + (1.43 − 0.500i)37-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)3-s + (−0.433 − 0.900i)4-s + (0.294 − 0.955i)7-s + (−0.955 + 0.294i)9-s + (−0.826 + 0.563i)12-s + (−0.997 + 0.0747i)13-s + (−0.623 + 0.781i)16-s + (−1.70 − 0.457i)19-s + (−0.988 − 0.149i)21-s + (−0.294 − 0.955i)25-s + (0.433 + 0.900i)27-s + (−0.988 + 0.149i)28-s + (1.82 + 0.488i)31-s + (0.680 + 0.733i)36-s + (1.43 − 0.500i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.976 - 0.216i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.976 - 0.216i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6201042213\)
\(L(\frac12)\) \(\approx\) \(0.6201042213\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.149 + 0.988i)T \)
7 \( 1 + (-0.294 + 0.955i)T \)
13 \( 1 + (0.997 - 0.0747i)T \)
good2 \( 1 + (0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.294 + 0.955i)T^{2} \)
11 \( 1 + (-0.563 + 0.826i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + (1.70 + 0.457i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.988 - 0.149i)T^{2} \)
31 \( 1 + (-1.82 - 0.488i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-1.43 + 0.500i)T + (0.781 - 0.623i)T^{2} \)
41 \( 1 + (0.680 - 0.733i)T^{2} \)
43 \( 1 + (0.548 - 0.215i)T + (0.733 - 0.680i)T^{2} \)
47 \( 1 + (0.563 - 0.826i)T^{2} \)
53 \( 1 + (-0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.974 + 0.222i)T^{2} \)
61 \( 1 + (1.64 - 0.123i)T + (0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.216 - 0.0579i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.149 + 0.988i)T^{2} \)
73 \( 1 + (1.73 + 0.918i)T + (0.563 + 0.826i)T^{2} \)
79 \( 1 + (-0.433 + 0.751i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.433 - 0.900i)T^{2} \)
89 \( 1 + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.241 + 0.902i)T + (-0.866 + 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.857177210436208552281725843020, −8.156116180967920300099512592304, −7.36386589262847452049484690244, −6.49524142188700750255251870208, −6.04882160497446264901633820305, −4.74608161450532793598690898208, −4.40891932209757719852341080981, −2.69349855733416730872155226352, −1.69336640879571032071954416015, −0.45047638482190693969591226831, 2.35784046449394394281270542327, 3.10519422745805051960735440313, 4.28461334063412520248967784509, 4.70275304530923340427064502626, 5.68393098430927786156137137301, 6.52805598357093231853489021540, 7.81834497253582632878291628200, 8.324055652808701668952446694338, 9.062730546000729408612122890810, 9.670146061857479031401649253131

Graph of the $Z$-function along the critical line